In this paper, we have studied the DHW-formalism in the 1D electrostatic limit. It turns out that for this case, the 16 scalar equations of the general theory can be reduced to four scalar equations given in (REF ), which only needs to be complemented by Ampere's law (REF ). Systems similar to Eqs. (REF ) have been studied previously, e.g. by Ref. [1]}, who, however, did not include the dependence on perpendicular momentum. While a perpendicular momentum dependence was included in Ref. [2]}, this paper only studied the homogeneous limit. Also, none of these works treated the field self-consistently by simultaneously solving (REF ).
